Vector Partition Functions Research Articles

Applying computer algebra systems to study Chaundy-Bullard identities for the vector partition function with weight

An algorithm for obtaining the Chaundy-Bullard identity for a vector partition function with weight that uses computer algebra methods is proposed. To automate this process in Maple, an algorithm was developed and implemented that calculates the values of the vector partition function with weight by finding non-negative solutions of systems of linear Diophantine equations that are used to form the identities involved. The algorithm’s input data is represented by the set of integer vectors that form a pointed lattice cone and by some point from this cone, and the Chaundy-Bullard identity for the vector partition function with weight is its output. The code involved is stored in the depository and is ready-to-use. An example demonstrating the algorithm’s operation is given.

Applying Computer Algebra Systems to Study Chaundy-Bullard Identities for the Vector Partition Function with Weight

Applying Computer Algebra Systems to Study Chaundy-Bullard Identities for the Vector Partition Function with Weight

Vector partition functions and Kronecker coefficients

The Kronecker coefficients are the structure constants for the restriction of irreducible representations of the general linear group GL(nm) into irreducibles for the subgroup GL(n) × GL(m). In this work we study the quasipolynomial nature of the Kronecker function using elementary tools from polyhedral geometry. We write the Kronecker function in terms of coefficients of a vector partition function. This allows us to define a new family of coefficients, the atomic Kronecker coefficients. Our derivation is explicit and self-contained, and gives a new exact formula and an upper bound for the Kronecker coefficients in the first nontrivial case.

Counting Integer Points of Flow Polytopes

The Baldoni–Vergne volume and Ehrhart polynomial formulas for flow polytopes are significant in at least two ways. On one hand, these formulas are in terms of Kostant partition functions, connecting flow polytopes to this classical vector partition function, fundamental in representation theory. On the other hand, the Ehrhart polynomials can be read off from the volume functions of flow polytopes. The latter is remarkable since the leading term of the Ehrhart polynomial of an integer polytope is its volume! Baldoni and Vergne proved these formulas via residues. To reveal the geometry of these formulas, the second author and Morales gave a fully geometric proof for the volume formula and a partial generating function proof for the Ehrhart polynomial formula. The goal of the present paper is to provide a fully geometric proof for the Ehrhart polynomial formula for flow polytopes.

Graded Betti numbers of powers of ideals

Using the concept of vector partition functions, we investigate the asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field. Our main results state that if the polynomial ring is equipped with a positive ℤ d -grading, then the Betti numbers of powers of ideals are encoded by finitely many polynomials. Specially, in the case of ℤ -grading, for each homological degree i we can split ℤ 2 = { ( μ , t ) ∣ t , μ ∈ ℤ } in a finite number of regions such that for each region there is a polynomial in μ and t that computes dim k ( Tor i S ( I t , k ) μ ) . This refines, in a graded situation, the result of Kodiyalam on Betti numbers of powers of ideals. Our main statement treats the case of a power products of homogeneous ideals in a ℤ d -graded algebra, for a positive grading.

Generating functions for vector partition functions and a basic recurrence relation

ABSTRACTWe define a generalized vector partition function and derive an identity for the generating series of such functions associated with solutions to basic recurrence relations of combinatorial analysis. As a consequence we obtain the generating function of the number of generalized lattice paths and a new version of the Chaundy-Bullard identity for the vector partition function.

Geometric Realizations and Duality for Dahmen–Micchelli Modules and De Concini–Procesi–Vergne Modules

We give an algebraic description of several modules and algebras related to the vector partition function, and we prove that they can be realized as the equivariant K-theory of some manifolds that have a nice combinatorial description. We also propose a more natural and general notion of duality between these modules, which corresponds to a Poincare duality-type correspondence for equivariant K-theory.

POWERS OF THE ETA-FUNCTION AND HECKE OPERATORS

Half-integer weight Hecke operators and their distinct properties play a major role in the theory surrounding partition numbers and Dedekind's eta-function. Generalizing the work of Ono in [K. Ono, The partition function and Hecke operators, Adv. Math.228 (2011) 527–534], here we obtain closed formulas for the Hecke images of all negative powers of the eta-function. These formulas are generated through the use of Faber polynomials. In addition, congruences for a large class of powers of Ramanujan's Delta-function are obtained in a corollary. We further exhibit a fast calculation for many large values of vector partition functions.

Vector partition functions and index of transversally elliptic operators

Let G be a torus acting linearly on a complex vector space M and let X be the list of weights of G in M. We determine the topological equivariant K-theory of the open subset M f of M consisting of points with finite stabilizers. We identify it to the space DM(X) of functions on the character lattice \( \widehat{G} \), satisfying the cocircuit difference equations associated to X, introduced by Dahmen and Micchelli in the context of the theory of splines in order to study vector partition functions (cf. [7]).This allows us to determine the range of the index map from G-transversally elliptic operators on M to generalized functions on G and to prove that the index map is an isomorphism on the image. This is a setting studied by Atiyah and Singer [1] which is in a sense universal for index computations.

Chopped and sliced cones and representations of Kac–Moody algebras

We introduce the notion of a chopped and sliced cone in combinatorial geometry and prove a structure theorem expressing the number of integral points in a slice of such a cone by means of a vector partition function. We observe that this notion applies to weight multiplicities of Kac–Moody algebras and to Clebsch–Gordan coefficients for semisimple Lie algebras. This has algorithmic applications, as we demonstrate computing some explicit examples.

Counting with rational generating functions

We examine two different ways of encoding a counting function: as a rational generating function and explicitly as a function (defined piecewise using the greatest integer function). We prove that, if the degree and number of input variables of the (quasi-polynomial) function are fixed, there is a polynomial time algorithm which converts between the two representations. Examples of such counting functions include Ehrhart quasi-polynomials, vector partition functions, integer points in parametric polytopes, and projections of the integer points in parametric polytopes. For this last example, this algorithm provides the first known way to compute the explicit function in polynomial time. We rely heavily on results by Barvinok and Pommersheim [Barvinok, A., Pommersheim, J., 1999. An algorithmic theory of lattice points in polyhedra. In: New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996–97). In: Math. Sci. Res. Inst. Publ., vol. 38. Cambridge Univ. Press, Cambridge, pp. 91–147], and also by Verdoolaege et al. [Verdoolaege, S., Seghir, R., Beyls, K., Loechner, V., Bruynooghe, M., 2007. Counting integer points in parametric polytopes using Barvinok’s rational functions, Algorithmica 48 (1), 37–66].

The Partial-Fractions Method for Counting Solutions to Integral Linear Systems

We present a new tool to compute the number $\phi_{\bf A} (b)$ of integer solutions to the linear system $$ x \geq 0, A x = b, $$ where the coefficients of $A$ and $b$ are integral. $\phi_{\bf A} (b)$ is often described as a vector partition function. Our methods use partial fraction expansions of Euler’s generating function for $\phi_{\bf A} (\b)$. A special class of vector partition functions are Ehrhart (quasi-)polynomials counting integer points in dilated polytopes.

A polynomiality property for Littlewood–Richardson coefficients

We present a polynomiality property of the Littlewood–Richardson coefficients c λμ ν . The coefficients are shown to be given by polynomials in λ, μ and ν on the cones of the chamber complex of a vector partition function. We give bounds on the degree of the polynomials depending on the maximum allowed number of parts of the partitions λ, μ and ν. We first express the Littlewood–Richardson coefficients as a vector partition function. We then define a hyperplane arrangement from Steinberg's formula, over whose regions the Littlewood–Richardson coefficients are given by polynomials, and relate this arrangement to the chamber complex of the partition function. As an easy consequence, we get a new proof of the fact that c Nλ Nμ Nν is given by a polynomial in N, which partially establishes the conjecture of King et al. (CRM Proceedings and Lecture Notes, Vol. 34, 2003) that c Nλ Nμ Nν is a polynomial in N with nonnegative rational coefficients.

Residue formulae for vector partitions and Euler–MacLaurin sums

Let V be an n-dimensional real vector space endowed with a rank-n lattice Γ . The dual lattice Γ ∗ = Hom(Γ.Z) is naturally a subset of the dual vector space V ∗. Let Φ = [β1, β2, . . . , βN ] be a sequence of not necessarily distinct elements of Γ ∗, which span V ∗ and lie entirely in an open halfspace of V ∗. In what follows, the order of elements in the sequence will not matter. The closed cone C(Φ) generated by the elements of Φ is an acute convex cone, divided into open conic chambers by the (n− 1)-dimensional cones generated by linearly independent (n−1)-tuples of elements of Φ . Denote by ZΦ the sublattice of Γ ∗ generated by Φ . Pick a vector a ∈ V ∗ in the cone C(Φ), and denote by ΠΦ(a) ⊂ R+ the convex polytope consisting of all solutions x = (x1, x2, . . . , xN) of the equation ∑Nk=1 xkβk = a in nonnegative real numbers xk . This is a closed convex polytope called the partition polytope associated to Φ and a. Conversely, any closed convex polytope can be realized as a partition polytope. If λ ∈ Γ ∗, then the vertices of the partition polytope ΠΦ(λ) have rational coordinates. We denote by ιΦ(λ) the number of points with integral coordinates in ΠΦ(λ). Thus ιΦ(λ) is the number of solutions of the equation ∑N k=1 xkβk = λ in nonnegative integers xk . The function λ → ιΦ(λ) is called the vector partition function associated to Φ . Obviously, ιΦ(λ) vanishes if λ does not belong to C(Φ) ∩ZΦ .

Residue formulae, vector partition functions and lattice points in rational polytopes

We obtain residue formulae for certain functions of several variables. As an application, we obtain closed formulae for vector partition functions and for their continuous analogs. They imply an Euler-MacLaurin summation formula for vector partition functions, and for rational convex polytopes as well: we express the sum of values of a polynomial function at all lattice points of a rational convex polytope in terms of the variation of the integral of the function over the deformed polytope.

On vector partition functions

We present a structure theorem for vector partition functions. The proof rests on a formula due to Peter McMullen for counting lattice points in rational convex polytopes.

Geometry and combinatorics related to vector partition functions

Geometry and combinatorics related to vector partition functions

Vector partitions and combinatorial identities

In this note we show how certain relations between vector partition functions can be deduced from certain identities. A relation connecting vector partitions having odd components and those having distinct parts will be proved. A combinatorial proof of Jacobi's Identity similar to Franklin's proof of Euler identity is suggested. The last section includes numerical values of Pr(n, n) and qr(n, m). These results suggest the unique maxima property of Pr(n, m) for fixed n, m and r varying. In the Jacobi Identity